Abstract. We propose a novel method, called spherical authalic energy minimization (SAEM), for computing spherical area-preserving parameterizations of genus-zero closed surfaces with strong theoretical foundations. The global convergence of the associated computational algorithm is theoretically guaranteed. In addition, we introduce a Riemannian bijective correction method that ensures the bijectivity of the resulting mapping under mild assumptions. Numerical experiments show that SAEM effectively minimizes area distortion and achieves bijective mappings, outperforming state-of-the-art methods. Finally, we demonstrate the practical utility of SAEM in shape description.

Abstract. Surface parameterization is a fundamental concept in fields such as differential geometry and computer graphics. It involves mapping a surface in three-dimensional space onto a two-dimensional parameter space. This process allows for the systematic representation and manipulation of surfaces of complicated shapes by simplifying them into a manageable planar domain. In this paper, we propose a new iterative algorithm for computing the parameterization of simply connected open surfaces that achieves an optimal balance between angle and area distortions. We rigorously prove the global convergence of the iteration in our algorithm. Numerical experiments demonstrate that the resulting mappings are bijective and effectively balance angular and area accuracy across various triangular meshes. Additionally, we present the practical usefulness of the proposed algorithm by applying it to represent surfaces as geometry images.

Abstract. Volumetric stretch energy has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models [Formula: see text]. However, this approach still lacks theoretical support. In this paper, we provide a theoretical foundation for volumetric stretch energy minimization (VSEM) to show that a map is a precise volume-/mass-preserving parameterization from [Formula: see text] to a region of a specified shape if and only if its volumetric stretch energy reaches [Formula: see text], where [Formula: see text] is the total mass of [Formula: see text]. We use VSEM to compute an [Formula: see text]-volume-/mass-preserving map [Formula: see text] from [Formula: see text] to a unit ball, where [Formula: see text] is the gap between the energy of [Formula: see text] and [Formula: see text]. In addition, we prove the efficiency of the VSEM algorithm with guaranteed asymptotic R-linear convergence. Furthermore, based on the VSEM algorithm, we propose a projected gradient method for the computation of the [Formula: see text]-volume-/mass-preserving optimal mass transport map with a guaranteed convergence rate of [Formula: see text], and combined with Nesterov-based acceleration, the guaranteed convergence rate becomes [Formula: see text]. Numerical experiments are presented to justify the theoretical convergence behavior for various examples drawn from known benchmark models. Moreover, these numerical experiments show the effectiveness of the proposed algorithm, particularly in the processing of 3D medical MRI brain images.

Abstract. The stretch energy is a fully nonlinear energy functional that has been applied to the numerical computation of area-preserving mappings. However, this approach lacks theoretical support and the analysis is complicated due to the full nonlinearity of the functional. In this paper, we establish a theoretical foundation of stretch energy minimization (SEM) for the computation of area-preserving mappings: the sufficient and necessary conditions for the energy minimizers are mappings being area-preserving. In addition, we derive a neat gradient formula of the functional and develop the associated line search gradient descent method of SEM with theoretically guaranteed convergence. Also, a simple post-processing technique is developed to guarantee the bijectivity of produced mappings. Furthermore, we generalized the theoretical results to the stretch energy for arbitrary area measures and the balanced energy so that the mass-preserving and distortion-balancing mappings can also be computed by minimizing the generalized stretch energy and the balanced energy, respectively. Numerical experiments and comparisons to another state-of-the-art algorithm are demonstrated to validate the effectiveness, accuracy, and robustness of SEM for computing area-preserving mappings.

Surface conformal parameterizations have been widely applied to various tasks in computer graphics. In this paper, we develop a convergent conformal energy minimization (CCEM) iterative algorithm via the line-search gradient descent method with a quadratic approximation for the computation of disk-shaped conformal parameterizations of simply connected open triangular meshes. In addition, we prove the global convergence of the proposed CCEM iterative algorithm. Moreover, under some mild assumptions, we prove the existence of a nontrivial solution, which is a local minimum of the conformal energy with a bijective boundary map. The numerical experiments indicate that the efficiency of the proposed CCEM algorithm is greatly improved and that the accuracy is competitive with that of state-of-the-art algorithms.

Parameterizations of manifolds are widely applied to the fields of numerical partial differential equations and computer graphics. To this end, in recent years several efficient and reliable numerical algorithms have been developed by different research groups for the computation of triangular and tetrahedral mesh parameterizations. However, it is still challenging when the topology of manifolds is nontrivial, e.g., the 3-manifold of a topological solid torus. In this paper, we propose a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of toroidal polyhedra with a single boundary being mapped to a standard torus. In addition, the algorithm can also be used to compute the equiareal mapping between a genus-one closed surface and the standard torus. Numerical experiments indicate that the developed algorithm is effective and performs well on the bijectivity of the mapping. Applications on manifold registrations and partitions are demonstrated to show the robustness of our algorithms.

Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher-resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected $3$-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle- and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.